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Can someone show what the trajectory of an object falling on a "pancake-shape" Earth would be, according to the gravitation law? Also if Earth is flat but not infinite, it must have edges. Since nobody has found any "edges" on Earth, if the Earth is flat, either it must be infinite in all directions (easy to disprove), or it has a very special shape with a surface that only has one side, like a Klein bottle (see picture) so that edges do not exist. But even if this was the case (I am joking here) how would gravitation work on such a planet? Anyone interested in doing simulations or solve the differential equations, to show the trajectory of an object falling on a pancake or Klein bottle, assuming the only force at play is gravitation?

By the way, and this is a bit more difficult to disprove, there are still people who believe that Earth, despite being a sphere, is shallow, with a big empty hole inside. Even fewer people believe that depending on where you are in the universe, gravitation could be negative or positive. 

Do you know other myths about how our universe is structured?

Klein Bottle

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You can rule out the Klein bottle; first, it is not flat; second, in a 3d space it is self-piercing, which would be hard not to notice.

A better option is to propose that the Earth is flat and finite, but it spins so fast that the periphery of the disc moves almost at the speed of light, making getting there infinitely hard. 

I found a Stack Exchange question that presents the math for a cylindrical object's gravitational attraction:

https://physics.stackexchange.com/questions/382574/gravitational-fo...

The "pancake" shape would fit this formula, being an extremely oblate cylinder.

I'd offer more than just the link, but pure math (beyond 2nd-semester calc) was always a weak spot for me.

There are some obvious areas in which the flat earth hypothesis falls apart and which have been known since ancient times, and which are readily verifiable by lay observers:

1.  The phenomenon of the horizon.  Even over "flat" terrain (such as wide bodies of water), one can only see so far.  And instead of objects simply becoming smaller and smaller as distance increases until they are no longer visible, they appear to sink below the horizon, even when viewed in a telescope.  Unless the speed of light were much slower than is generally supposed (which would open up a completely different can of worms), one should be able to look all the way across even the widest oceans, were the earth flat.

2.  The apparent positions of the sun, moon, and stars at different latitudes.  It is well known that the stars visible from New York are different than those visible in Sydney, yet in each case, they appear to rotate around a different point (the north and south celestial poles).  If the earth were flat, there would only be one.  The phenomenon of day and night being of different lengths at different latitudes (and more different from each other the further one gets from the Equator) is also only explainable if the earth is globular.  Likewise, the phenomenon of the midnight sun in the north and south polar zones is only explainable if the earth is globular.

3.  (Related to 1)  At higher elevations, one can see farther, even if there are no obstructions.  Simple geometry explains this if the earth is a globe, but not if it is a disk or a plane.

4.  The time of day (ie. the apparent position of the sun in the sky) varies with longitude.  This was not easy to verify in the days before modern telecommunications, but is readily verifiable today.

Wondering also how you could explain (if Earth was flat) the fact that water flushed from a bathtub 10 meters North of the equator,  circles in the opposite direction than the same bathtub 10 meters South of the equator. I've heard that there is actually a tourist attraction somewhere in Africa featuring this curiosity. Not sure if it is related to the rotation of Earth (Coriolis effect?) and if the Earth needs to be a sphere for this to happen, but that's one way to prove that (1) equator exists, (2) you can locate its exact position with rudimentary technology (bathtubs) and (3) you can walk or sail along that "line" and eventually, after 40,000 kilometers, arrive back to your starting point (so that "line" must be a curve.) Anyway, flat-Earth believers probably believe that Earth does not rotate, and rather, the Sun, Milky Way and all other celestial structures rotate around the Earth.  

The same mechanism applies to hurricanes as well, rotating clockwise or anti-clockwise depending on the hemisphere.

John L. Ries said:

There are some obvious areas in which the flat earth hypothesis falls apart and which have been known since ancient times, and which are readily verifiable by lay observers:

1.  The phenomenon of the horizon.  Even over "flat" terrain (such as wide bodies of water), one can only see so far.  And instead of objects simply becoming smaller and smaller as distance increases until they are no longer visible, they appear to sink below the horizon, even when viewed in a telescope.  Unless the speed of light were much slower than is generally supposed (which would open up a completely different can of worms), one should be able to look all the way across even the widest oceans, were the earth flat.

2.  The apparent positions of the sun, moon, and stars at different latitudes.  It is well known that the stars visible from New York are different than those visible in Sydney, yet in each case, they appear to rotate around a different point (the north and south celestial poles).  If the earth were flat, there would only be one.  The phenomenon of day and night being of different lengths at different latitudes (and more different from each other the further one gets from the Equator) is also only explainable if the earth is globular.  Likewise, the phenomenon of the midnight sun in the north and south polar zones is only explainable if the earth is globular.

3.  (Related to 1)  At higher elevations, one can see farther, even if there are no obstructions.  Simple geometry explains this if the earth is a globe, but not if it is a disk or a plane.

4.  The time of day (ie. the apparent position of the sun in the sky) varies with longitude.  This was not easy to verify in the days before modern telecommunications, but is readily verifiable today.

The draining-water swirl isn't relevant, for two reasons:

1) Coriolis acceleration works in 2D.  If you can find a playground that still has a 'merry-go-round' toy, have two kids on opposite sides, get the merry-go-round up to a decent rotational rate, then have one of them (try to) throw a ball to the other one.  In the thrower's perspective the ball will seem to veer off at a sharp curve instead of travelling straight to the kid on the other side.

2) In my senior-year Aerospace classes (ca 1986), when we covered the Coriolis acceleration, the professor addressed the water-swirl business and explained that it was a longstanding urban legend.  He'd done the calculations himself and established that the acceleration was far too small to overcome the inertia of the water.  There's a current Stack Exchange article which likewise affirms this: 

https://physics.stackexchange.com/questions/7738/why-theres-a-whirl...

I've heard from a missionary friend about the tourist attraction (apparently, the local children are happy to demonstrate with a bucket of water and some bird feathers, for a pretty nominal amount of money). But the math & physics show that it's merely a staged show.  (The spirit of P.T. Barnum lives on, even in places that never heard of him.)  Likely the kids have learned the knack for giving the bucket a strategic slosh-twist to get the appropriate spin going.

Oh, one other point from that Aerospace class, which actually IS relevant:

The professor I mentioned, also described an authentic instance of Coriolis from the days of "wooden ships & iron men."  Although the folks of the time didn't fully understand it, they discovered by experience that when firing ship's cannons at an enemy, the cannon shot would 'drift' from a straight line by some margin, based on the direction you fired and how far away the target was.  They actually had charts which laid out out in detail how to adjust the aim accordingly.  And those carefully-calculated charts served them quite well... UNTIL the first time they tried to use the data while fighting battles in the Southern hemisphere.  Suddenly all the adjustments were reversed!  Makes perfect sense once you understand the physics and know that the earth is a globe.  But very hard to explain otherwise.

Thanks Jim for your insightful comments. Very interesting, and obviously, you know better than me.

The draining water, I thought, was a function of conservation of angular momentum. The water has to start out still relative to the tub, but of course you, the tub, and still water are in motion.

https://www.wired.com/2014/05/wuwt-foucaults-pendulum/ The Foucault's pendulum shows the Earth is rotating. What calculations would be needed to show angular momentum on a sphere vs a flat whatever this nut job thinks the Earth is shaped as? Theorize what measurements would be seen on a flat disc and then show actual measurements agree with a spherical shape.

Shadows from sticks. Think Sun dials.

https://www.space.com/38931-kids-can-prove-earth-round.html

One of the best documented methods for determining the Earth's roundness was first performed (to our knowledge) by the ancient Greeks. This was achieved by comparing the shadows of sticks in different locations. When the sun was directly overhead in one place, the stick there cast no shadow. At the same time in a city around 500 miles north, the stick there did cast a shadow.

If the Earth were flat then both sticks should show the same shadow (or lack of) because they would be positioned at the same angle towards the sun. The ancient Greeks found the shadows were different because the Earth was curved and so the sticks were at different angles. They then used the difference in these angles to calculate the circumference of the Earth. They managed to get it to within 10% of the true value – not bad for around 250 B.C.

Yes, the normal trigger for that drain swirl is 'remnant' motion of water in the container after it is filled.  If you wanted to see a genuine Coriolis-induced swirl, you'd need a container of absolutely still water and a very tiny drain hole.  Not to mention being at a relatively high latitude (much more than a few feet from the equator) -- the force vectors & trig functions work against you otherwise.  Ideal would be one of the poles, provided you don't end up with a bucket of ice before the experiment's done.  :)
 
Steve Bowling said:

The draining water, I thought, was a function of conservation of angular momentum. The water has to start out still relative to the tub, but of course you, the tub, and still water are in motion.

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